Physica D 241 (2012) 1660–1669 Contents lists available at SciVerse ScienceDirect Physica D journal homepage: www.elsevier.com/locate/physd Potential well metamorphosis of a pivoting fluid-filled container Si Mohamed Sah ⇤ , Brian P. Mann Department of Mechanical Engineering & Materials Science, Duke University, Durham, NC 27708, USA article info Article history: Received 7 January 2012 Received in revised form 21 June 2012 Accepted 3 July 2012 Available online 7 July 2012 Communicated by G. Stepan Keywords: Container Stability Bifurcation Experiment abstract This paper investigates the stability of a pivoting cylindrical container that is slowly filled with fluid. The action of filling the container with fluid causes the system’s potential energy to evolve and modify the stability of equilibria. We analyze the stability behavior of this system and find distinct regions where edge and spill conditions require alternative expressions for the system’s potential energy. The stability of the upright and tilt angle equilibria are studied using the Lagrange–Dirichlet theorem. We provide exact expressions for the potential energy of the system and bifurcation diagrams that compactly represent the stability behavior of the upright equilibria and additionally predict the presence of non-trivial or tilted equilibria. Theoretical investigations are then compared with a series of experimental tests that validate the container upright and tilted equilibria stability. © 2012 Elsevier B.V. All rights reserved. 1. Introduction Human sensory receptors provide a motivating example where nature has used the equilibrium of a fluid to perceive balance. As noted in Ref. [1], fluid filled cavities enable humans to detect their cranial position and to interpret acceleration. In Meteorology, a tipping bucket type of rain sensor is commonly used as a rain detector in electronic weather stations [2]. In this apparatus, rain collects into a self-emptying bucket which destabilizes and empties once the quantity of the collected water reaches a calibrated threshold; the number of times the bucket empties and resets is then counted and used to ascertain the amount of rainfall. Gardens are another area where a pivoting fluid-filled container appears. In addition to the spilling bucket fountains that are sometimes used for aesthetic or watering purposes, the Japanese culture has used a fountain called the Shishi Odoshi to scare away deer for centuries [3,4]. The present study of a fluid- filled cylinder also closely replicates the grasping of a drinking container with the human hand, where the thumb and forefinger grasp the container sides to create a pivot axis. Yet another application area for fluid-filled containers lies in the entertainment industry. For example, many theme parks use a self-emptying container to periodically dump large sums of water on willing participants at water parks—such as the ‘‘Tipping Tiki’’ in the Splash Lagoon Indoor Water Park, US. More complex ⇤ Correspondence to: Mechanical Engineering and Materials Science, 144 Hudson Hall, Duke University, Durham, NC 27708, USA. Tel.: +1 919 660 5348; fax: +1 919 660 8963. E-mail address: ss424@duke.edu (S.M. Sah). attractions have also used the transition from a stable to unstable equilibria to drive devices – such as the Germaul water automat at Helburn Palace, Austria – where the lower jaw, when it is filled with water, tips over and grabs a bent rod that actuates the tongue and the eyelids. Once emptied, the lower jaw returns to its closed position and the cycle is repeated as the water is poured into the lower jaw [5,6]. The primary connection amongst the above examples is the interaction of a fluid and rigid structure to bring about transitions in stability. Another example related to the interaction of a fluid and rigid structure is the motion control of an open container with slosh. Sloshing is a phenomenon that describes the fluid motion in partially filled containers. This phenomenon can cause the fluid to spill out of the container. Therefore controlling the motion of open containers in order to minimize the sloshing is of great interest—especially in the packaging industry [7,8]. Various works have appeared on the stability of containers that are immersed in a fluid. For example, Douglas studied the stability of a floating cylinder and found that for certain values of the length–diameter ratio, the cylinder attains stable flotation with its axis at one particular inclination between the horizontal and vertical [9]. Delbourgo [10] investigated the stability of different configurations of a floating plank with rectangular cross section. In another work Erdös and Schilber [11] examined the equilibrium configuration of solid prisms of squares and equilateral triangular cross section floating in a liquid. Closer to the present paper, Trahan and Kalmár-Nagy [12] studied the case of a rectangular liquid-filled vessel. They demonstrated the co-existence of several equilibrium configurations of the vessel. The equilibrium states were found by deriving the position vector of the center of gravity of the liquid, and the potential function was used to determine the stability of 0167-2789/$ – see front matter © 2012 Elsevier B.V. All rights reserved. doi:10.1016/j.physd.2012.07.001